HERMITIANS IN MATRIX ALGEBRAS WITH OPERATOR NORM – II

We continue our investigation of the real space H of Hermitian matrices in $${M_n}(\mathbb{C})$$ with respect to norms on $${\mathbb{C}^n}$$ . We complete the commutative case by showing that any proper real subspace of the real diagonal matrices on $${\mathbb{C}^n}$$ can appear as H. For the non-commutative case, we give a complete solution when n=3 and we provide various illustrative examples for n ≥ 4. We end with a short list of problems.

Connected Fundamental Groups and Homotopy Contacts in Fibered Topological (C, R) Space

The algebraic as well as geometric topological constructions of manifold embeddings and homotopy offer interesting insights about spaces and symmetry. This paper proposes the construction of 2-quasinormed variants of locally dense p-normed 2-spheres within a non-uniformly scalable quasinormed topological (C, R) space. The fibered space is dense and the 2-spheres are equivalent to the category of 3-dimensional manifolds or three-manifolds with simply connected boundary surfaces. However, the disjoint and proper embeddings of covering three-manifolds within the convex subspaces generates separations of p-normed 2-spheres. The 2-quasinormed variants of p-normed 2-spheres are compact and path-connected varieties within the dense space. The path-connection is further extended by introducing the concept of bi-connectedness, preserving Urysohn separation of closed subspaces. The local fundamental groups are constructed from the discrete variety of path-homotopies, which are interior to the respective 2-spheres. The simple connected boundaries of p-normed 2-spheres generate finite and countable sets of homotopy contacts of the fundamental groups. Interestingly, a compact fibre can prepare a homotopy loop in the fundamental group within the fibered topological (C, R) space. It is shown that the holomorphic condition is a requirement in the topological (C, R) space to preserve a convex path-component. However, the topological projections of p-normed 2-spheres on the disjoint holomorphic complex subspaces retain the path-connection property irrespective of the projective points on real subspace. The local fundamental groups of discrete-loop variety support the formation of a homotopically Hausdorff (C, R) space.

On the Topological Structure and Properties of Multidimensional (C, R) Space

Generally, the linear topological spaces successfully generate Tychonoff product topology in lower dimensions. This paper proposes the construction and analysis of a multidimensional topological space based on the Cartesian product of complex and real spaces in continua. The geometry of the resulting space includes a real plane with planar rotational symmetry. The basis of topological space contains cylindrical open sets. The projection of a cylindrically symmetric continuous function in the topological space onto a complex planar subspace maintains surjectivity. The proposed construction shows that there are two projective topological subspaces admitting non-uniform scaling, where the complex subspace scales at a higher order than the real subspace generating a quasinormed space. Furthermore, the space can be equipped with commutative and finite translations on complex and real subspaces. The complex subspace containing the origin of real subspace supports associativity under finite translation and multiplication operations in a combination. The analysis of the formation of a multidimensional topological group in the space requires first-order translation in complex subspace, where the identity element is located on real plane in the space. Moreover, the complex translation of identity element is restricted within the corresponding real plane. The topological projections support additive group structures in real one-dimensional as well as two-dimensional complex subspaces. Furthermore, a multiplicative group is formed in the real projective space. The topological properties, such as the compactness and homeomorphism of subspaces under various combinations of projections and translations, are analyzed. It is considered that the complex subspace is holomorphic in nature.

Hyperbolicity of the complement of arrangements of non complex lines

The goal of this paper is twofold. We study holomorphic curves f:C ? C3 avoiding four complex hyperplanes and a real subspace of real dimension five in C3 where we study the cases where the projection of f into the complex projective space CP2 is constant. On the other hand, we investigate the kobayashi hyperbolicity of the complement of five perturbed lines in CP2.

On logarithmic residue of monogenic functions in a three-dimensional commutative algebra with one-dimensional radical

We consider monogenic functions taking values in a three-dimensional commutative algebra A2 over the field of complex numbers with one- dimensional radical. We calculate the logarithmic residues of monogenic functions acting from a three-dimensional real subspace of A2 into A2. It is shown that the logarithmic residue depends not only on zeros and singular points of a function but also on points at which the function takes values in ideals of A2, and, in general case, is a hypercomplex number.

A wall-crossing formula for degrees of Real central projections

The main result is a wall-crossing formula for central projections defined on submanifolds of a Real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the degree depends on the projection is a new phenomenon, specific to Real algebraic geometry. We illustrate this phenomenon in many interesting situations. The crucial assumption on the class of maps we consider is relative orientability, a condition which allows us to define a ℤ-valued degree map in a coherent way. We end the article with several examples, e.g. the pole placement map associated with a quotient, the Wronski map, and a new version of the Real subspace problem.

On the classification of the real vector subspaces of a quaternionic vector space

We prove the classification of the real vector subspaces of a quaternionic vector space by using a covariant functor which associates, to any pair formed of a quaternionic vector space and a real subspace, a coherent sheaf over the sphere.

A note on good real perturbations of singularities

One can study the local topology of images of complex analytic maps and the restriction of such a map to a real subspace. Conditions are given for the topology of the image multiple point spaces of the complex and real maps to have the same homology or homotopy type.

COMPLEX FENCHEL-NIELSEN COORDINATES FOR QUASI-FUCHSIAN STRUCTURES

Let Fg be a closed oriented surface of genus g ≥ 2 and let [Formula: see text] be the space of marked quasi-fuchsian structures on Fg. Let [Formula: see text] be a set of non-intersecting, non-trivial simple closed curves on Fg that cuts Fg into pairs of pants components. In this note, we construct global complex coordinates for [Formula: see text] relative to [Formula: see text] giving an embedding of [Formula: see text] into [Formula: see text]. The totally real subspace of [Formula: see text] with respect to these coordinates is the Teichmüller Space [Formula: see text] of marked hyperbolic structures on Fg, the coordinates reduce to the usual Fenchel-Nielsen coordinates for [Formula: see text] relative to [Formula: see text]. Various properties of the embedded image are studied.